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Van Vleck Temperature-Independent Paramagnetism

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Van Vleck Temperature-Independent Paramagnetism Van Vleck lecturing in Sterling Hall Attendees at the Sixth Solvay Conference,1930. Van Vleck is in the back row, third from the right, standing next to Enrico Fermi. Albert Einstein is seated in the front row, fifth from the right. J. H. Van Vleck and Magnetism at the University of Wisconsin: 1928 ‐1934 Susceptibilities Local Fields Van Vleck Susceptibility Papers: 1928-1929 1)On Dielectric Constants and Magnetic Susceptibilities in the New Quantum Mechanics, Part III, Phys. Rev. 31, 587 (1928). (Minnesota) 2) The Effect of Second Order Zeeman Terms on Magnetic Susceptibilities in the Rare Earth and Iron Groups, Phys. Rev. 34, 1494 (1929) (with Amelia Frank). Paramagnetic Susceptibilities M = χH χ > 0 Free ion, LS coupling Zeeman interaction: gJμBJzH 2 2 χ= (1/3)gJ μB J(J+1)/kT Temperature-dependent! Van Vleck Temperature-Independent Paramagnetism Excited State ex Δ <ex|HZee|g> <g|HZee|ex> Ground state g Zeeman interaction: HZee = μBH(Lz+2Sz) Van Vleck Paramagnetism (con’t) Magnetic field admixes excited state wave function into ground state and ground state into excited state: ψg’= ψg − (<ex|HZee|g>/Δ)ψex ψex’= ψex + (<g|HZee|ex>/Δ) ψg Usual case: kT << Δ. Only ground state occupied. Temperature-independent contribution to susceptibility: 2 2 δχ = 2NμB |<ex|(Lz+2Sz)|g>| /Δ Ion La+++ Ce+++ Pr+++ Nd+++ Pm+++ Sm+++ Eu+++ Gd+++ (2S+1) 1 2 3 4 5 6 7 8 LJ S0 F5/2 H4 I9/2 I4 H5/2 F0 S7/2 Old 0 2.54 3.58 3.62 2.68 0.84 0 7.9 V V‐ 0 2.56 3.62 3.69 2.87 1.83 3.56 7.9 Frank Expt. 0 2.39 3.60 3.62 _____ 1.54 3.61 8.2 Cabrera Expt. 0 ______ 3.47 3.51 _____ 1.32 3.12 8.1 Meyer Apparent Bohr magneton numbers for first half of rare earth group 2 1/2 Apparent magneton number = [3kTχexp(T)/NμB ] Table from Van Vleck and Frank, 1929 Crystal Field Effects (Crystalline Stark Effect) Van Vleck-Frank paper dealt with free ions. Effects of interactions between magnetic ion and neighboring non-magnetic ions? H. Bethe, Ann. d. Physik 3, 133 (1929) Susceptibility Papers: 1932 1) The Influence of Crystalline Fields on the Susceptibilities of Salts of Paramagnetic Ions. I.The Rare Earths, Especially Pr, and Nd, W. G. Penney and R. Schlapp, Phys. Rev. 41, 194 (1932). (Van Vleck post-docs!) 2) Theory of the Variations in Paramagnetic Anisotropy Among Different Salts of the Iron Group, J. H. Van Vleck, Phys. Rev. 41, 208 (1932). (follows Penney-Schlapp paper) 3) The Influence of Crystalline Fields on the Susceptibilities of Salts of Paramagnetic ions. II. The Iron Group, Especially Ni, Cr, and Co, R. Schlapp and W. G. Penney, Phys. Rev. 42, 666 (1932). (companion to Van Vleck paper) Paramagnetic Anisotropy in Iron Group Salts: Ni++ vs Co++ 1) Ni++ has nearly isotropic susceptibility with a ‘spin only’ Curie constant 2) Co++ has significant anisotropy in susceptibility 3) Ni++ and Co++ are adjacent in periodic table and in orbital F states Van Veck Explanation: Crystal Field Effects 1) Reversal of crystal field levels d83F (Ni++) → d74F (Co++) Ni++ Co++ 2) Quenching of ground state orbital magnetic moment (Ni++) The Theory of Electric and Magnetic Susceptibilities J. H. Van Vleck Professor of Theoretical Physics in The University of Wisconsin Oxford University Press 1932 (manuscript completed in 1931) Notes for a Second Edition (given to Chun Lin by Abigail Van Vleck after John’s death) Sections of the Local Field Chapter 25.Depolarizing (or Demagnetizing) Corrections 26.The Lorentz Field and its 4π/3 Catastrophe 27.The Onsager Local Field 28.Dipole Interaction Treated by Statistical Mechanics 29.Short Range Order – Kirkwood’s Formula 30.Limits of Validity of the Clausius-Mossotti Formula – The Translational Fluctuation Effect 31.Inadequacy of the Local Field Concept 32. Incipient Saturation Effects Local Field “The effective average field to which a molecule is subjected when a macroscopic field E is applied” J. H. Van Vleck (from The Theory of Electric and Magnetic Susceptibilities) Phenomenological Electrostatic Theories for the Local Field Lorentz (1878) eLOC = E +(4π/3)P = [(2 + ε)/3]E Onsager (1936) eLOC =[3ε/(2ε + 1)]E (polar molecules) eLOC =[(2 + ε)/3]E (polarizable molecules) (polar) J. H. Van Vleck, J. Chem. Phys. 5, 320 (1937) Which is the better approximation? Van Vleck’s Footnote “Apparently the first attempt to treat dipolar coupling in dielectrics by means of statistical mechanics, using this interaction in the partition function instead of employing electromagnetic theory, was made by the writer (J. Chem. Phys. 5, 320, and especially, 556 (1937)). The analysis in the present section follows to a considerable extent that in this earlier work but is presented in improved and somewhat more condensed form.” Van Vleck’s Analysis Microscopic calculation of the dielectric constant for a liquid with molecules that are both polar and polarizable (α≠0). (1) Expand the partition function to second order in the dipolar interaction. (2) Calculate the dielectric constant. (3) Use the Clausius-Mossotti (C-M) equation as the basis for comparison with Lorentz-Debye and Onsager theories. C-M Eq. (ε−1)/(ε +2) = (4πN/3V)[α + ?] Conclusion According to Van Vleck, “the results of [this] calculation are thus about half way between those of the Onsager and [Lorentz-]Debye models.” Edited Chapter IV is available as a pdf from the UW-Physics web site http://www.physics.wisc.edu/vanvleck/ Van Vleck Biographical Memoirs P. W. Anderson, National Academy of Sciences, 1987 B. Bleaney, Fellows of the Royal Society, 1982 Van Vleck’s Local Field Analysis (con’t) Lorentz-Debye: (ε−1)/(ε +2) = (4πN/3V)(α + μ2/3kT) Onsager: (ε−1)/(ε +2) = (4πN/3V)(α + 3εγ(μ2/kT)/[(ε+2)(2ε+1)]) α = polarizability, μ = dipole moment, γ = f(α, ε).
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